Consider the following matrix equation:
\begin{equation} \operatorname{diag}(AXA^\dagger)=0 \end{equation} where $\operatorname{diag}(.)$ represent the diagonal elements. with $X$ being the variable matrix and $A$ being an arbitrary matrix with constant elements and $A^\dagger=(A^*)^T$. Assuming they both have complex entries, what is the sufficient and necessary condition that the answer to the above equation be \begin{equation} X=0 \end{equation} Update: if there is not a necessary and sufficient condition, is there a necessary condition?
meaning is there a condition on entries of $A$ such that $\operatorname{diag}(AXA^\dagger)=0$ implies $X=0$ ?
Your notation is very confusing. I suppose that $A^\dagger$ means $A^\ast=\overline{A}^{\,T}$ (the conjugate transpose of $A$) rather than $(A^\ast)^T=\overline{A}$ (the complex conjugate of $A$). In this case, the statement $$ \forall X,\ \operatorname{diag}(AXA^\dagger)=0\Rightarrow X=0\tag{1} $$ holds if and only if $A$ is a nonzero column vector.
Let $A$ be $m\times n$. When $n>1$, $f:X\mapsto \operatorname{diag}(AXA^\dagger)$ is a linear map from $M_n(\mathbb C)$ to $\mathbb C^n$. Since $\dim M_n(\mathbb C)=n^2>n=\dim\mathbb C^n$, $\ker f$ is always nonzero regardless of the value of $A$.
When $n=1$, $X$ is a scalar and $\operatorname{diag}(AXA^\dagger)=X(|a_1|^2,|a_2|^2,\ldots,|a_m|^2)^\top$. Therefore $(1)$ holds if and only if $A\ne0$.